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Holonomy theory of Finsler manifolds. (English) Zbl 1392.53079
Falcone, Giovanni (ed.), Lie groups, differential equations, and geometry. Advances and surveys. Cham: Springer; Palermo: Università degli Studi di Palermo (ISBN 978-3-319-62180-7/hbk; 978-3-319-62181-4/ebook). UNIPA Springer Series, 265-320 (2017).
The holonomy group of a Finsler manifold is the group generated by parallel translations along loops with respect to the canonical connection.
If the Riemannian holonomy groups have been extensively studied and completely classified, for the Finslerian holonomy only few results are known and many of these are different from the Riemannian one. The authors of this article have seriously studied this subject and obtained fundamental results on it. Since 2011, they have published six other articles on the holonomy theory of Finsler manifolds. They have introduced a method for the investigation of holonomy properties of non-Riemannian Finsler manifolds by constructing tangent Lie algebras to the holonomy group: the curvature algebra, the infinitesimal holonomy algebra, and holonomy algebra.
In the present book chapter, the authors present this method and give a unified treatment of their results. In particular, they show that the dimension of these tangent algebras is usually greater than the possible dimensions of Riemannian holonomy groups and in many cases is infinite. It is proved that the holonomy group of a locally projectively flat Finsler manifold of constant curvature is finite-dimensional if and only if it is a Riemannian manifold or a flat Finsler manifold. Then the authors prove that the topological closure of the holonomy group of a certain class of simply connected, projectively flat Finsler 2-manifolds of constant curvature is not a finite-dimensional Lie group, and that its topological closure is the connected component of the full diffeomorphism group of the circle.
The article has the following sections. 1. Introduction, 2. Preliminaries, 3. Diffeomorphism groups and their tangent algebras, 4. Curvature algebra, 5. Holonomy algebra, 6. Infinite dimensional subalgebras of the infinitesimal holonomy algebra, 7. Dimension of the holonomy group, 8. Maximal holonomy.
The bibliography is quasi-exhaustive, including old articles [G. Randers, Phys. Rev., II. Ser. 59, 195–199 (1941; Zbl 0027.18101); C. Ehresmann, in: Centre Belge Rech. Math., Colloque Topologie, Bruxelles, du 5 au 8 juin 1950, 29–55 (1951; Zbl 0054.07201); A. Borel and A. Lichnerowicz, C. R. Acad. Sci., Paris 234, 1835–1837 (1952; Zbl 0046.39801)], until recent work by L. Kozman and by the authors.
For the entire collection see [Zbl 1381.53010].
Reviewer: Ioan Pop (Iaşi)

MSC:
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C29 Issues of holonomy in differential geometry
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